Ok I was able to get the hot water turned off and changed out my angle valve with a new one. I also noticed that the nipple coming out of the wall was in very bad shape too (plugged up with a bunch of black goo, and piece of the thread was chipped off), so I changed that as well. Turned out to be only a 10 minute job plus time spent running down the street to Ace Hardware.

I used threaded pipe compound on both ends of the nipple and made sure every thing is wrench tight. I still have that nagging drip from the faucet! I guess I am glad I changed out those older parts but what else could be the problem? The drip isn't as fast now that I changed out those parts but still a constant drip, just slower now.

Also do I need to be worried about all that black goo that was clogged in my hot water pipe? It didn't seem to be affecting the flow of the water or discolored the water.

Repairing or replacing the angle stop below the sink will not cure a dripping faucet, they are unrelated. The faucet has a bad washer or cartridge. The black goo does concern me since it could be bacteria.

Where would I find the cartridge on a faucet? I guess I don't know what that is.

As for the possibility of bacteria... how would I know for sure? Or how would I clean out that pipe run?